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Journal of Physics Condensed Matter

Autonomous topological time crystals and knotty molecular motors

Jin Dai, Xubiao Peng, Antti J. Niemi

Published Sep 9, 2020• 0 citations
Publisher

Abstract

We show that topology is a very effective tool, to construct classical Hamiltonian time crystals. For this we numerically analyze a general class of time crystalline Hamiltonians that are designed to model the dynamics of molecular closed strings. We demonstrate how the time crystalline qualities of a closed string are greatly enhanced when the string becomes knotted. The Hamiltonians that we investigate include a generalized Kratky-Porod wormlike chain model in combination with long range Coulomb and Lennard-Jones interactions. Such energy functions are commonplace in coarse grained molecular modeling. Thus we expect that physical realizations of Hamiltonian time crystals can be constructed in terms of knotted ring molecules.

DOI
10.1088/1361-648x/abb682

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We show that topology is a very effective tool, to construct classical Hamiltonian time crystals. For this we numerically analyze a general class of time crystalline Hamiltonians that are designed to model the dynamics of molecular closed strings. We demonstrate how the time crystalline qualities of a closed string are greatly enhanced when the string becomes knotted. The Hamiltonians that we investigate include a generalized Kratky-Porod wormlike chain model in combination with long range Coulomb and Lennard-Jones interactions. Such energy functions are commonplace in coarse grained molecular modeling. Thus we expect that physical realizations of Hamiltonian time crystals can be constructed in terms of knotted ring molecules.
A time crystal is a time dependent physical system that does not reach a standstill, even in state of minimum energy. Here we show that the stability of a time crystal can be enhanced by its topology. For this we simulate time crystals made of chainlike ensembles of mutually interacting point particles. When we tie the chain into a knot we find that its timecrystalline qualities improve. The theoretical models we consider are widely used in coarse grained descriptions of linear polymers. Thus we expect that physical realizations of time crystals can be found in terms of knotted molecules.
Time crystals correspond to a phase of matter where time-translational symmetry (TTS) is broken. Up to date, they are well studied in open quantum systems, where an external drive allows us to break discrete TTS, ultimately leading to Floquet time crystals. At the same time, genuine time crystals for closed quantum systems are believed to be impossible. In this study we propose a form of a Hamiltonian for which the unitary dynamics exhibits the time crystalline behavior and breaks continuous TTS. This is based on a spin-1/2 many-body Hamiltonian which has long-range multispin interactions in the form of spin strings, thus bypassing previously known no-go theorems. We show that quantum time crystals are stable to local perturbations at zero temperature. Finally, we reveal the intrinsic connection between continuous and discrete TTS, thus linking the two realms.
Classical Hamiltonian time crystals are constrained energy minima that move along symmetry orbits at frequencies set by Lagrange multipliers. We construct quantum counterparts by minimizing the energy over density matrices with fixed spectrum and charge expectations. A symmetry-covariant quantum Landau--Lifshitz--Gilbert equation selects them as periodic attractors, preserving the spectrum and fixing the multipliers dynamically. A chiral spin trimer motivated by Cu3_33​ molecular magnets realizes this motion at a prepared magnetization, without a Zeeman term.
Time crystals are nonequilibrium phases of matter characterized by the emergence of temporal ordering, in which an interacting many-body system develops robust structure in its time evolution that is not trivially dictated by the external driving or environment. While related phenomena have long been studied in classical nonlinear systems, their realization in entangled quantum matter represents a distinct frontier. The theoretical understanding of discrete time crystals has substantially advanced, yet recent experiments using modern quantum devices and quantum processors reveal regimes beyond established paradigms. These developments call for an extended classification of time-crystalline phases according to both their stabilization mechanisms and their physical character, including discrete and continuous, closed and open, critical, topological, quasiperiodic, and controlled realizations. We review recent implementations of time crystals on quantum platforms and propose such a classification framework, identifying promising directions for the discovery of novel time-crystalline phases of matter.
We introduce a new class of out-of-equilibrium noninteracting topological phases: the topological space-time crystals. These are time-dependent quantum systems that do not have discrete spatial translation symmetries but instead are invariant under discrete space-time translations. Similar to the Floquet-Bloch systems, the space-time crystals can be described by a frequency-domain-enlarged Hamiltonian, which is used to classify topologically distinct space-time crystals. We show that these space-time crystals can be engineered from conventional crystals with an additional time-dependent drive that behaves like a traveling wave moving across the crystal. Interestingly, we are able to construct 1D and 2D examples of topological space-time crystals based on tight-binding models that involve only one orbital, in contrast to the two-orbital minimal models for any previously discovered static or Floquet topological phases with crystalline structures.
We explore a self-operating, room-temperature 1024-channel quantum computer that fundamentally shifts computing by replacing qubits and logic gates with polyatomic time crystals (PTCs), six-dimensional topological objects. This architecture is built upon 16 supramolecular organic gels that serve as customizable quantum circuits, driven by homology–cohomology duality engines that autonomously resolve topological inconsistencies in unknown data to build responses as an automaton with <327 680 variables in 1024 channels. The system uses eight symmetry protocols to construct a topological version of the input as PTCs; frequency-combed, tri-phase quantum light reads the output PTC, and magnetic perturbations control conversion of 6D PTCs into 12D PTC outputs encoded as soliton-polariton condensates of long-lived triplets flowing through the gel networks. The goal is to solve quantum Merlin–Arthur-hard and bounded-error quantum polynomial time-class problems by offloading computational complexity to topology and quantum dynamics as a tree tensor network (TTN), enabling the computer to discover hidden rules in fast-changing big data with high fault tolerance. Intractability is attacked not by speeding up a logical algorithm, but by offloading combinatorics to the topology of quantum dynamics: the exponential solution space becomes a landscape of knot states and degeneracy groups, and the “answer” is the TTN itself, forming a globally coherent, mass-carrying cluster. PTCs are the TTN medium where decisions follow compatibility of three topology layers—local singularity bonding, collective shape metamorphosis, and asymptotic feedback knots—rather than any finite logical circuit. This lets PTC hardware extract deeply non-local, history-dependent structures in data as genuine topological phase decisions, giving extraordinary physical and computational significance.